2009 | OriginalPaper | Chapter
The Spectral Theorem -I
Author : Rajendra Bhatia
Published in: Notes on Functional Analysis
Publisher: Hindustan Book Agency
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Let A be a Hermitian operator on the space ℂn. Then there exists an orthonormal basis {e j } of ℂn each of whose elements is an eigenvector of A. We thus have the representation (24.1)<math display='block'> <mrow> <mi>A</mi><mo>=</mo><mstyle displaystyle='true'> <munderover> <mo>∑</mo> <mrow> <mi>j</mi><mo>=</mo><mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <msub> <mi>λ</mi> <mi>j</mi> </msub> <mrow><mo>〈</mo> <mrow> <mo>⋅</mo><mo>,</mo><msub> <mi>e</mi> <mi>j</mi> </msub> </mrow> <mo>〉</mo></mrow><msub> <mi>e</mi> <mi>j</mi> </msub> <mo>,</mo> </mrow> </mstyle> </mrow> </math>$$A = \sum\limits_{j = 1}^n {{\lambda _j}\left\langle { \cdot ,{e_j}} \right\rangle {e_j},}$$ where Ae j = λ j e j . We can express this in other ways. Let λ1 > λ2 > ⋯ > λ k be the distinct eigenvalues of A and let m1, m2,&, m k be their multiplicities. Then there exists a unitary operator U such that (24.2)<math display='block'> <mrow> <mi>U</mi><mo>*</mo><mi>A</mi><mi>U</mi><mo>=</mo><mstyle displaystyle='true'> <munderover> <mo>∑</mo> <mrow> <mi>j</mi><mo>=</mo><mn>1</mn> </mrow> <mi>k</mi> </munderover> <mrow> <msub> <mi>λ</mi> <mi>j</mi> </msub> <msub> <mi>P</mi> <mi>j</mi> </msub> <mo>,</mo> </mrow> </mstyle> </mrow> </math>$$U*AU = \sum\limits_{j = 1}^k {{\lambda _j}{P_j},}$$, where P1, P2,&, P k are mutually orthogonal projections and (24.3)<math display='block'> <mrow> <mstyle displaystyle='true'> <munderover> <mo>∑</mo> <mrow> <mi>j</mi><mo>=</mo><mn>1</mn> </mrow> <mi>k</mi> </munderover> <mrow> <msub> <mi>P</mi> <mi>j</mi> </msub> <mo>=</mo><mi>I</mi><mo>.</mo> </mrow> </mstyle> </mrow> </math>$$\sum\limits_{j = 1}^k {{P_j} = I.}$$. The range of P j , is the m j -dimensional eigenspace of A corresponding to the eigenvalue λ j . This is called the spectral theorem for finite-dimensional operators.